Archive for April, 2011

Flavor: Translating a proof from one context to another.

April 30, 2011

1. What’s going on mathematically?

Sometimes after working through a proof in a paper, it’s useful to try to translate the proof from that context to another. {As a concrete example, I was recently reading Bousfield’s construction of a complementation operator a(-) on the object level of the category of spectra, with the goal of using ideas from his proofs to construct such an operator in the derived category D(R) of a ring.}

2. What is the emotional and logistical context?

To do this well, I have to be in top mathematical form, ready to pull on all the resources at my disposal. So I wait until the emotional and logistical context is most conducive – i.e. quiet, still, spacious, rested, relaxed, happy, perhaps caffeinated.

3. What thoughts are there?

Usually you don’t just have to “translate” the proof, but also the statement of the claim. This quickly becomes a jumble of cause and effect, of constantly adjusting hypotheses and consequences. {What should be the definition of a(E)? Well, it depends on what kind of constructions I can do, to get nice cofibers. The obvious one, given by the definition of cellular complexes, apparently wasn’t going to work (I came up with a counterexample to be sure),  but maybe I could tweak that filtration, or start with it to build a different one.} This takes maximum mental bandwidth. It also helps to be in good communication with my intuition on the subject (whatever you take that to mean), since intuition is an excellent guide for this.

At the same time, to use the old proof in the new context requires not just a logical understanding that the original proof works, but a robust grasp on why it works, and how. What is really going on {i.e. what key aspects of spectra is Bousfield using}? What proof techniques are at play? {In this case, it was transfinite induction.}

4. What quality of awareness?

My mind has to be relaxed and open, to start with. What seems crucial to this sloppy driving around in the muck, is being able to hold out on smaller details, to suspend disbelief and pursue consequences quickly and often. It’s like my mind simultaneously throws six or seven balls in the air at once, and my job is to keep them all bouncing around for a little while. The fact that I’m working off of a concrete original proof, sort of keeps the balls contained within four walls, and gives me a solid point of reference and interface.

There’s very little room for self-reflection or distraction, or else I’ll miss some potentially fertile combination of the ideas. This is a very absorbed state of mind.

5. What emotions?

It feels very spontaneous, and helps to be feeling flexible and willing. I don’t know how the proof will translate, what it might allow me to prove or disprove in the new context, so there’s uncertainty and excitement. Sometimes I get anxious that the proof won’t translate to prove such and such; or that someone smarter than me, with a better grasp on the material, better intuition, or wider knowledge, would be able to translate successfully but I won’t. It helps to confront this anxiety head-on, since it is very unconducive to the flexibility and intensity required to do the work.

6. What does it resolve to, after how much time?

This can resolve in two ways. On the one hand, I may land on some result, and succeed in having extracted something useful out of the original proof’s ideas and techniques. This result, however small, can be built upon and will suggest new directions, of course. Or else the experiment was unsuccessful, in which case there’s a reason it was unsuccessful, and I maybe can pursue that reason – maybe I need to read more about Y, or better understand X, or maybe it just won’t work.

7. How frequent is this flavor?

Frequent. It’s a great way of squeezing out new results. It’s really helpful to have the walls there when I’m trying to throw all those balls around at the same time.

8. What are good/bad ways to change or follow it up?

In my experience, it’s unhelpful to get too discouraged if things don’t go as I had hoped. It’s better to fill in some gaps, or take a different tack, and try again.


Connections:

In Proofs and Refutations, Imre Lakatos demonstrates the simultaneous co-creation of definitions, statements of claims, and proofs.

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Flavor: Stunned and sublime.

April 30, 2011

1. What’s going on mathematically?

After doing a lot of math, usually at least four hours, usually seated, I get up to go do something else.

2. What is the emotional and logistical context?

Usually I’ll be walking somewhere, or biking home. Sometimes it’s really late at night.

3. What thoughts are there?

Not many, and none about math.

4. What quality of awareness?

I become hyper-aware of my physical body and sensations – of the wind on my skin, the air and my breath, of sounds and movement, colors, or visual patterns. There is only a little processing of the sensations, for the most part it feels like I’ve just returned from the rarefied cloud of math to return into the living, breathing world, and that world’s forms are flooding over me with no resistance.

5. What emotions?

There’s a deep peace and calmness of body and thoughts, a happy glowing, and a gentle feeling that everything is going to be okay. A lot of times I’ll catch myself in a wide-eyed stare, that brings with it a pressure behind my eyes that I associate with the experience.

6. What does it resolve to, after how much time?

It usually goes away naturally, after 10-15 minutes.

7. How frequent is this flavor?

It happens almost every time, for at least five minutes, after a math session of at least four hours. If I go for less than four hours, then I’ll still have some no-thought, physical relaxation and inner stillness, but without the deep resonating peacefulness, and without the sustained effect on my mode of perception.

8. What are good/bad ways to change or follow it up?

Good: enjoy it, be happy, and let it go. Bad: It’s a shame not to relish it.


Connections:

In my all-time favorite book, Hermann Hesse’s Magister Ludi (The Glass Bead Game), this experience, referred to as “cheerful serenity”, plays a big role.

Common Ground

April 28, 2011

The University of Washington’s Mathematics Department has an annual departmental potluck and musical event – a chance for faculty and grad students to socialize and show off some of their musical and performance talent. Each year I’ve tried to contribute something unique; last year in 2010 I wanted to entertain but also provoke thought. The opportunity to perform in front of a captive audience of my colleagues and mentors is too tempting to resist.

My goal was to capture what it’s like to be a mathematician, in an entertaining yet self-reflective way. My hypothesis is that, across our different fields, the combinatorialist, analyst, and algebraic topologist, etc, share many day-to-day and week-to-week experiences. Of course there is a diversity of experiences, but there are also similarities. It’s rare for the department to come together, and I wanted to use the occasion as an opportunity to express and address these commonalities.

For example, there’s the common, yet somewhat contradictory, philosophical stance of the practicing mathematician – “a formalist during the week and a Platonist on Sunday” (Hersh and Davis). There’s the nonlinear feedback loop and co-evolution of definitions, hypotheses, conjectures, and theorems (as in Lakatos’ Proofs and Refutations). There’s getting into the one-pointed concentration zone, where you feel like a mere conduit for the mathematical flow to express itself. There’s the use of machinery to break problems into smaller, more manageable pieces, slowly extracting small results that bit by bit add together. There’s getting completely stuck. There’s the paperwork and grading, and the extremely difficult casual conversation with the non-mathematician about your work. There are the times that distance and leisure suggest new insights or new perspectives. There’s the creating of new perspectives, and the discovery of what must be so. There’s that moment when the last piece falls into place, and certainty resonates within you (à la Poincaré). There’s the disappointment of finding a counterexample, and having to throw out a week (or month) of work.

My performance tried to express all these, in under ten minutes. It was part theater, part performance art, with a dynamic soundtrack and lots of bizarre props. There’s a video of this performance on my website, www.forthelukeofmath.com (well, any day now).

In a sense, this blog is an attempt at continuing the programme set out in that performance: to express some of the common experiences encountered along the mathematical path.

After the show, the feedback among my classmates and the faculty was better than I hoped. “Luke, you hit the nail on the head!”, “We were all just talking about it together… that’s exactly what it’s like to do math!” “The music you chose was perfect!”

I’ve tried to engage other mathematicians, in more or less formal settings, in this level of meta-mathematics. A continuing issue is that most mathematicians feel that any discussion involving math must proceed with utmost rigor and objectivity. I strongly disagree with this sentiment.  My performance at the math musicale succeeded, I think, because the medium – absurdist performance art – made it clear: we are contemplating mathematics, but in a looser, more subjective way.

So, as you read the blog, please withhold your insistence on rigor. It is necessary to math, but it is not all of math. Try to be mindful of how it feels to surrender rigor – how the subjectivity floods in, and in the flood what are the branches that we can hold on to? These are the threads of intersubjectivity – shared experience.

An everyday metaphor

When you go to the movies to see a romantic drama, the plot usually pulls from a database of common themes.  Likewise when you listen to the lyrics of songs on the radio.  If you’re watching a romance or listening to the radio, you sort of know what to expect.  The themes can be trite and arranged uninterestingly, as in your average romantic comedy.  They can also be presented artfully and uniquely, as in a Leonard Cohen ballad.  These themes are, in part, normative scripts that we are socialized to (e.g. hetero- versus homo-relationship paradigms), but they also express universals that appear across all or most human cultures.

So if this blog were about partnerships rather than mathematics, some flavors might be: the second date, meeting the family, cooking a meal together.  Some seasons might be: getting to know each other, traveling together, long-distance.  Thank Goodness I’m not making a blog about those things.

Flavor: Working through a proof.

April 28, 2011

1. What’s going on mathematically?

Trying to understand a theorem, and the proof techniques used.

2. What is the emotional and logistical context?

This is relatively straightforward. It takes concentration, but the concentration sort of builds on itself.  It’s like reading a really engaging story – you get roped in. So it has to be pretty quiet and peaceful, but my attention is not that frail and I’ll get into “the zone” pretty quickly. It’s a good one for when I feel like feeling like a mathematician, but have some procrastination/anxiety/staleness.  So I may start out slow, but pretty quickly feel happy. (I tried to do this yesterday, after rock climbing all day, and while eating dinner, getting devoured by mozzies, and sitting in an uncomfortable chair, but I failed.  After a good night’s sleep and breakfast and coffee, it was effortless.) I must do it with a pad of paper, to scribble out drawings, diagrams, and recapitulations (i.e. I rewrite the sentence on my own, to somehow  concretize every single word).

3. What thoughts are there?

It’s in no sense easy, but is easier than some other flavors; because I get so roped in, it’s easy to proceed.  But it really demands that you harness all your knowledge about the concepts on hand – every sentence usually has two or three clauses, each of which requires an intense “Why is this true?”  To figure this out, I need to draw on things I learned a long time ago, or recently, or vague connections between ideas, and often look up definitions or other results.  When I can’t figure out steps, I keep track of those failures, or come up with various provisional explanations, whose correctness I can usually establish by the end of the proof, using the test of consistency.  Some leaps are simply too great, and I quickly give up hope of understanding them; I used to ask my advisor about all of these technicalities, and sometimes still do.  In the end, I (hopefully) can find a simple conceptual organization (“what’s really going on?”, “what’s the essence?”, “what’s the shape of the proof?”) within the proof, quite different from the original narrative.

4. What quality of awareness?

Quite single-pointed and dialogical, narrative.  It’s very intimate and microscopic, tracing the smallest steps of logic and understanding, at a nice slow pace; it’s not about the destination, but the process; any understanding is good understanding, and fortunately the ideas often resonate beautifully.  The provisional explanations take a wider focus, but usually are born and die without too much trouble and time.

5. What emotions?

Happy and content; peaceful.  Especially when I can take my time with the ideas, and enjoy them.  I get frustrated when I can’t figure out parts, and annoyed when there are flippant comments that I have no choice but to take on faith (…sometimes these are even wrong!).

6. What does it resolve to, after how much time?

It usually spurs on more non-linear work, pretty quickly.

7. How frequent is this flavor?

It seems healthy to work through a proof every once in a while, since there are so many helpful clues hidden in the proof techniques of theorems.  It’s probably the most helpful part of figuring out my research – giving me the highest density of new research ideas – and so I should do it as often as I can.

8. What are good/bad ways to change or follow it up?

Good: dive right into applying the proof techniques.  Bad: get really frustrated at not understanding or forgetting things.


Connections:

This blog post by Terence Tao has links to several comments on how to read mathematics.

Season: Entering a Field.

April 27, 2011

1. What mathematical activities? What level of rigor?

This experience lasted from the time I began reading courses with my soon-to-be PhD advisor, until right before my General Exam. My activities were divided into two main types.

First, I was loosely trying to map out the new terrain. It was like someone dropped me in a new city, with a bicycle, and said, “You have a week to map out the whole city.” In a non-rigorous, and sometimes cursory way, I was rapidly traversing whatever conceptual avenues presented themselves to me. Over time, the same words and references kept popping up, and I built a rough map of what ideas were important nodes, what papers or people were central. It was helpful to build a historical/chronological map of the ideas, in addition to my content-based map. This exploration was done largely online, but also through talking with professors and other grad students.

Second, I was endeavoring to slowly, methodically build a solid foundation of understanding. In an extremely rigorous, pedantic way I inched my way through book after book, paper after paper. I tried to do every exercise, tried to understand every clause of every proof. Each field has its own collection of common proof techniques and tricks, and it’s necessary to get a functional grasp of these tools, add them to your toolbox.

2. What relevant interactions with other mathematicians?

Frequent advisor meetings were important, for pointed questions on proofs (“How does he get from here to here?”), as well as vague intuition-building ponderings (“How important is it to localize at p?”). Other professors and grad students were great at filling in a picture of what matters, what people care about, what’s really going on.

3. How does it feel, what is the mood?

This was a fun and exciting time, since I enjoyed the field (algebraic topology) that I was entering (see #5 below). In the first case, it felt like I was an explorer in a new country, trying to understand the history, landscape, important people and places. In the second case, I was starting the foundation for a new mathematical castle, and was finding the material understandable and aesthetically beautiful. Having a solid, comprehensive understanding of any subject is an intellectual’s dream.

4. What state of mind? stable vs. chaotic? focused vs. dispersed?

In the first case, my mind was very chaotic, mercurial, dispersed, and outwardly oriented. My goal was breadth and big-picture understanding, which is like flying above the forest with nowhere to land. Nothing really made sense, I was just exposing myself to what was out there and trying to get used to it, trying to accept whatever pathways and features were laid out below me. This was unsettling and in some ways very shallow and unsatisfying.

But I complemented this with the second approach, of depth and rigor. My mind was like a frog or snail, slowly coming to understand a few trees in the forest.

5. What type of self-reflection during the experience, and did it help?

I recognized these two very different approaches, and how they were complementing each other. This allowed me to intentionally play them off of each other. Reading history and mapping the social network of algebraic topologists would unnerve me as being shallow and “un-mathematical”, and I would switch to crawling through some proof or exercise. If this began to feel hopelessly specialized, I would switch back to a broader perspective – and I would find that studying one tree had helped me to speak the language of the whole forest, and now my broader study was yielding more fruit. Back and forth, between breadth and depth, using self-reflection to decide when it was time to switch. (As we said on the Appalachian Trail, “walk until you feel like stopping, then stop until you feel like walking.”) If math is too broad and shallow, or math is too deep and specific, it leaves a bad taste in your awareness.

I think the shallower, more holistic type of understanding is often under-appreciated by mathematicians, or at least no one taught me that I should intentionally pursue a big-picture map. (On the contrary, a professor once told me, “The thing about you, Luke, is that you’re not really a mathematician, you’re more like a spectator of mathematics. You’re really enthusiastic about knowing all the stats of all the players, but you never get out on field.”)  Self-reflection showed me this prejudice about what it means to “understand,” and helped me catch myself when I was getting unnecessarily unnerved.

6. An everyday metaphor for the experience?

This season was very much like the first weeks and months of a new relationship. There’s lots of getting to know each other, lots of cans of worms that need to be  opened one-by-one and dealt with, lots of snippets and intuitions that need to be gradually pulled together to form a holistic map. There are both chronological and as-you-are-today maps to be sketched. Shallow, yet broad, shared experiences complement those that are deeper and more focused.

7. An example of a good day and a bad day?

On a good day I would make progress on some paper or book I was reading, and then sit down and have a Wikipedia wandering session, and find that I was recognizing more and more of the words that showed up. On a bad day, I would get stuck on a proof and be too stubborn to give up, or would procrastinate facing the rolling-up-of-the-sleeves-anxiety that follows me everywhere.

8. What did you do when you were stuck?

I would switch between the two approaches (breadth and depth), and wait until my next advisor meeting to move forward.

9. When and why did it end?

In a sense, you never really stop learning about your field. But the initial culture-shock phase of mapping out a foreign land has ended, as has the original foundation building. I do occasionally visit neighboring cities and try to map them out, or methodically add a new story or wing to my rendition of the algebraic topology castle.

sticky post: Flavors and What?

April 27, 2011

The purpose of this blog is to document the flavors and seasons of the mathematical experience.  I’d like to dedicate the blog to the discoverer/inventer of projective geometry, Girard Desargues.

His 1639 masterpiece, “Rough draft for an essay on the results of taking plane sections of a cone,” with only 50 copies published, was ignored for about 200 years. According to  Raymond Wilder, there are two possible explanations for why his work, “one of the most unsuccessful great works ever published,” went unappreciated for so long.

1.  Perhaps it wasn’t until the mid 1800’s that the mathematical community was prepared for the revolutionary idea that Euclidean geometry was just one of many equally valid and self-consistent geometries.  (In the 1830’s, the incomparable Gauss developed his own non-Euclidean ideas, but chose to carry them to his grave (1855), because, as he wrote to a friend, “I fear the ‘clamor of the Boeotians.'”)

2. Or maybe it’s because Desargues’ work was “couched in a strange terminology, much of which was borrowed from botany.”

For more info about this project, go to the About the project page.

Find a list of all flavors and seasons in the Table of Contents.  Recent posts are below.  Please add comments.